Namespaces
Variants
Actions

Difference between revisions of "Solenoid"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Arc (topology)|arc)
m (tex encoded by computer)
 
Line 1: Line 1:
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s0860401.png" /> be a sequence of positive integers. From <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s0860402.png" /> one constructs a topological space as follows.
+
<!--
 +
s0860401.png
 +
$#A+1 = 29 n = 0
 +
$#C+1 = 29 : ~/encyclopedia/old_files/data/S086/S.0806040 Solenoid
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s0860403.png" /> be a torus in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s0860404.png" />; inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s0860405.png" /> one takes a torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s0860406.png" /> wrapped around longitudinally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s0860407.png" /> times, in a smooth fashion without folding back; inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s0860408.png" /> one takes a torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s0860409.png" /> wrapped around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604010.png" /> times in the same way. Continuing this procedure indefinitely, one obtains a decreasing sequence of tori. Its intersection is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604012.png" />-adic solenoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604013.png" />.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
The basic properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604014.png" /> are that it is a one-dimensional [[Continuum|continuum]] which, moreover, is indecomposable (cf. [[Indecomposable continuum|Indecomposable continuum]]).
+
Let  $  \mathbf n = \langle  n _ {i} \rangle _ {i} $
 +
be a sequence of positive integers. From  $  \mathbf n $
 +
one constructs a topological space as follows.
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604015.png" /> is also a [[Topological group|topological group]]; this can be seen if one considers an alternative construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604016.png" /> as the inverse limit of the following inverse sequence:
+
Let  $  T _ {0} $
 +
be a torus in  $  \mathbf R  ^ {3} $;
 +
inside  $  T _ {0} $
 +
one takes a torus  $  T _ {1} $
 +
wrapped around longitudinally  $  n _ {1} $
 +
times, in a smooth fashion without folding back; inside  $  T _ {1} $
 +
one takes a torus  $  T _ {2} $
 +
wrapped around  $  n _ {2} $
 +
times in the same way. Continuing this procedure indefinitely, one obtains a decreasing sequence of tori. Its intersection is called the  $  \mathbf n $-
 +
adic solenoid  $  \Sigma _ {\mathbf n }  $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604017.png" /></td> </tr></table>
+
The basic properties of  $  \Sigma _ {\mathbf n }  $
 +
are that it is a one-dimensional [[Continuum|continuum]] which, moreover, is indecomposable (cf. [[Indecomposable continuum|Indecomposable continuum]]).
  
where each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604018.png" /> is the unit circle and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604019.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604020.png" />. There are various other ways in which one can construct the solenoids, see, e.g., [[#References|[a3]]].
+
$  \Sigma _ {\mathbf n }  $
 +
is also a [[Topological group|topological group]]; this can be seen if one considers an alternative construction of  $  \Sigma _ {\mathbf n }  $
 +
as the inverse limit of the following inverse sequence:
  
Solenoids were first defined by L. Vietoris [[#References|[a2]]] (for the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604021.png" />) and by D. van Dantzig [[#References|[a1]]] (for all constant sequences).
+
$$
 +
{} \dots S _ {3}  \rightarrow ^ { {f _ 3} }  S _ {2}  \mathop \rightarrow \limits ^ { {f _ {2} }}  S _ {1}  \rightarrow ^ { {f _ 1} }  S _ {0} ,
 +
$$
 +
 
 +
where each  $  S _ {i} $
 +
is the unit circle and  $  f _ {i} :  S _ {i} \rightarrow S _ {i-} 1 $
 +
is defined by  $  f _ {i} ( z)= z ^ {n _ {i} } $.
 +
There are various other ways in which one can construct the solenoids, see, e.g., [[#References|[a3]]].
 +
 
 +
Solenoids were first defined by L. Vietoris [[#References|[a2]]] (for the sequence $  \langle  2, 2 ,\dots \rangle $)  
 +
and by D. van Dantzig [[#References|[a1]]] (for all constant sequences).
  
 
Solenoids are also important in [[Topological dynamics|topological dynamics]]; on them one can define a [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] structure [[#References|[a4]]] which has a locally disconnected minimal set of almost-periodic motions.
 
Solenoids are also important in [[Topological dynamics|topological dynamics]]; on them one can define a [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] structure [[#References|[a4]]] which has a locally disconnected minimal set of almost-periodic motions.
  
There is a complete classification of the solenoids: first, without loss of generality one may assume that the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604022.png" /> are prime. Call two sequences of primes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604024.png" /> equivalent if one can delete from each a finite number of terms such that in the reduced sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604026.png" /> every prime is counted the same number of times. One can then show that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604028.png" /> are homeomorphic if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086040/s08604030.png" /> are equivalent. See [[#References|[a5]]] and [[#References|[a6]]].
+
There is a complete classification of the solenoids: first, without loss of generality one may assume that the numbers $  n _ {i} $
 +
are prime. Call two sequences of primes $  \mathbf p $
 +
and $  \mathbf q $
 +
equivalent if one can delete from each a finite number of terms such that in the reduced sequences $  \mathbf p  ^  \prime  $
 +
and $  \mathbf q  ^  \prime  $
 +
every prime is counted the same number of times. One can then show that $  \Sigma _ {\mathbf p }  $
 +
and $  \Sigma _ {\mathbf q }  $
 +
are homeomorphic if and only if $  \mathbf p $
 +
and $  \mathbf q $
 +
are equivalent. See [[#References|[a5]]] and [[#References|[a6]]].
  
 
Finally one can characterize the solenoids as those metric continua that are homogeneous and have the property that every proper subcontinuum is an [[Arc (topology)|arc]]. See [[#References|[a7]]].
 
Finally one can characterize the solenoids as those metric continua that are homogeneous and have the property that every proper subcontinuum is an [[Arc (topology)|arc]]. See [[#References|[a7]]].

Latest revision as of 08:14, 6 June 2020


Let $ \mathbf n = \langle n _ {i} \rangle _ {i} $ be a sequence of positive integers. From $ \mathbf n $ one constructs a topological space as follows.

Let $ T _ {0} $ be a torus in $ \mathbf R ^ {3} $; inside $ T _ {0} $ one takes a torus $ T _ {1} $ wrapped around longitudinally $ n _ {1} $ times, in a smooth fashion without folding back; inside $ T _ {1} $ one takes a torus $ T _ {2} $ wrapped around $ n _ {2} $ times in the same way. Continuing this procedure indefinitely, one obtains a decreasing sequence of tori. Its intersection is called the $ \mathbf n $- adic solenoid $ \Sigma _ {\mathbf n } $.

The basic properties of $ \Sigma _ {\mathbf n } $ are that it is a one-dimensional continuum which, moreover, is indecomposable (cf. Indecomposable continuum).

$ \Sigma _ {\mathbf n } $ is also a topological group; this can be seen if one considers an alternative construction of $ \Sigma _ {\mathbf n } $ as the inverse limit of the following inverse sequence:

$$ {} \dots S _ {3} \rightarrow ^ { {f _ 3} } S _ {2} \mathop \rightarrow \limits ^ { {f _ {2} }} S _ {1} \rightarrow ^ { {f _ 1} } S _ {0} , $$

where each $ S _ {i} $ is the unit circle and $ f _ {i} : S _ {i} \rightarrow S _ {i-} 1 $ is defined by $ f _ {i} ( z)= z ^ {n _ {i} } $. There are various other ways in which one can construct the solenoids, see, e.g., [a3].

Solenoids were first defined by L. Vietoris [a2] (for the sequence $ \langle 2, 2 ,\dots \rangle $) and by D. van Dantzig [a1] (for all constant sequences).

Solenoids are also important in topological dynamics; on them one can define a flow (continuous-time dynamical system) structure [a4] which has a locally disconnected minimal set of almost-periodic motions.

There is a complete classification of the solenoids: first, without loss of generality one may assume that the numbers $ n _ {i} $ are prime. Call two sequences of primes $ \mathbf p $ and $ \mathbf q $ equivalent if one can delete from each a finite number of terms such that in the reduced sequences $ \mathbf p ^ \prime $ and $ \mathbf q ^ \prime $ every prime is counted the same number of times. One can then show that $ \Sigma _ {\mathbf p } $ and $ \Sigma _ {\mathbf q } $ are homeomorphic if and only if $ \mathbf p $ and $ \mathbf q $ are equivalent. See [a5] and [a6].

Finally one can characterize the solenoids as those metric continua that are homogeneous and have the property that every proper subcontinuum is an arc. See [a7].

References

[a1] D. van Dantzig, "Ueber topologisch homogene Kontinua" Fund. Math. , 15 (1930) pp. 102–125
[a2] L. Vietoris, "Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen" Math. Ann. , 97 (1927) pp. 454–472
[a3] E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 1 , Springer (1979)
[a4] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)
[a5] R.H. Bing, "A simple closed curve is the only homogeneous bounded plane continuum that contains an arc" Canad. Math. J. , 12 (1960) pp. 209–230
[a6] M.C. McCord, "Inverse limit sequences with covering maps" Trans. Amer. Math. Soc. , 114 (1965) pp. 197–209
[a7] C.L. Hagopian, "A characterization of solenoids" Pacific J. Math. , 68 (1977) pp. 425–435
How to Cite This Entry:
Solenoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Solenoid&oldid=48745